SAMPLE EXAM
MATH 200 Calculus 1 (Bueler)
Final Exam: Comprehensive Part
100 points total for this part. You have 120 minutes for both parts together.
1.
(b)
2.
(b)
(a)
(5 pts) Compute f 0 (5) if f (x) = x−2 .
(5 pts) Use the definition of the derivative to compute f 0 (5) if f (x) = x−2 .
(a)
(5 pts) Find dy/dx if
√
y = 2 x + 6ex
(5 pts) Compute
Z
√
2 x + 6ex dx
3. (10 pts) The equation
x2 − xy + y 2 = 4
is an ellipse. (2, 0) is a point on this ellipse. Find the equation of the tangent line to this
ellipse at this point. (Hint: There is no need to sketch any graph.)
4. Consider the curve
y=
4−x
.
x−1
(a)
(5 pts) Is this curve even, odd, or neither?
(b)
(5 pts) Find the intercepts (locations where the graph crosses the x and the y axes).
(c)
(5 pts) Find the intervals on which the curve is increasing and decreasing.
(d)
(5 pts) Sketch the graph, showing any asymptotes.
5. (10 pts) Find the absolute maximum and minimum of f (t) = 2t − tan t on the interval
[0, π/4].
2
Final Exam: Comprehensive Part CONTINUED
SAMPLE EXAM
6. (a) (5 pts) Bismuth-210 is a radioactive substance whose mass decays exponentially.
It has a half-life of 5.0 days. A sample originally has a mass of 800 mg. Find a formula for
the mass y(t) remaining after t days.
(b)
(5 pts) Find the mass remaining after 15 days.
7. (10 pts) Define
“ lim f (x) = L ”
x→a
You may use either the sentence definition written many times in lecture, or you may use
the –δ definition.
8. (10 pts) Compute
Z 7
cos x
4 dx
3 sin x
9. (10 pts) Find the area between y = ex and y = 1/x on the interval [1, 2].
Extra Credit. (3 pts) Show that
d
1
(arcsec x) = √
dx
x x2 − 1
You may use known derivatives of trignometric functions, but not, naturally, the derivatives
of inverse trigonometric functions. You may use standard trigonometric identities.